Trees with unique minimum p-dominating sets

Abstract

Let p be a positive integer and G = (V(G),E(G)) a simple graph. A p-dominating set of G is a subset S of V(G) such that every vertex not in S is dominated by at least p vertices in S. The p-domination number γ_P(G) is the minimum cardinality among the p-dominating sets of G. In this paper, for p ≥ 2, we give three equivalent conditions for trees with unique minimum p-dominating sets and also give a constructive characterization of such trees.